Source: Journal of Generalized Lie Theory and Applications, Volume 9, Number 1, 5 pages.
Abstract:
Given an associative graded algebra equipped with a degree $+1$ differential
Δ we define an $A_\infty$-structure that measures the failure of Δ
to be a derivation. This can be seen as a non-commutative analog of generalized
BValgebras. In that spirit we introduce a notion of associative order for the
operator Δ and prove that it satisfies properties similar to the
commutative case. In particular when it has associative order 2 the new product
is a strictly associative product of degree +1 and there is compatibility
between the products, similar to ordinary BV-algebras. We consider several
examples of structures obtained in this way. In particular we obtain an
$A_\infty$-structure on the bar complex of an $A_\infty$-algebra that is
strictly associative if the original algebra is strictly associative. We also
introduce strictly associative degree $+1$ products for any degree $+1$ action
on a graded algebra. Moreover, an $A_\infty$-structure is constructed on the
Hochschild cocomplex of an associative algebra with a non-degenerate inner
product by using Connes’ B-operator.